Summarising Data

12.6 Summarising data (EMG74)

After data has been collected, classified and organised it is not always possible to mention every piece of data in a report. Instead we summarise data by describing the whole data set using just a few numbers. Summarising data also makes it easier to analyse the data later.

Measures of central tendency and measures of spread (EMG75)

Data can be summarised by using measures of central tendency or measures of spread.

Measure of central tendency is a single value that attempts to show what the central position is of a set of data. Measures of spread describe how the data is spread out or dispersed.

There are three types of measures of central tendency: mean, mode and median.

Mean

The mean is the most common measure of central tendency that is used. It is also known as the average. It is calculated by adding all the values together and dividing by the number of values in the data set. E.g. If you have numbers \(\text{2}\); \(\text{6}\); \(\text{8}\); \(\text{10}\); \(\text{12}\); \(\text{14}\); \(\text{18}\); the mean is calculated as:

Worked example 2: Finding the mean

The frequency table below shows the test marks achieved by \(\text{20}\) learners. The test was marked out of \(\text{10}\). Calculate the mean mark.

Mark

\(\text{4}\)

\(\text{6}\)

\(\text{7}\)

\(\text{8}\)

\(\text{9}\)

\(\text{10}\)

Frequency

\(\text{2}\)

\(\text{4}\)

\(\text{3}\)

\(\text{6}\)

\(\text{3}\)

\(\text{2}\)

To find the mean we add all the values of the data and divide by how many data values there are. Therefore mean value of the numbers \(\text{4}\); \(\text{6}\); \(\text{7}\); \(\text{3}\); \(\text{4}\); \(\text{8}\); \(\text{4}\); \(\text{2}\); \(\text{9}\) is calculated as follows. We add \(\text{4}\) + \(\text{6}\) + \(\text{7}\) + \(\text{3}\) + \(\text{4}\) + \(\text{8}\) + \(\text{4}\) + \(\text{2}\) + \(\text{9}\) and get the following number \(\text{47}\). now we must count how many values we added together. In this example we used \(\text{9}\) values. So the mean is \(\frac{\text{47}}{\text{9}} = \text{5,2}\).

Here we need to add the marks of \(\text{20}\) learners, but because some of the marks are repeated we can use multiplication as a short method for adding the same number several times. The total of the marks = (\(\text{4}\) \(\times\) \(\text{2}\)) + (\(\text{6}\) \(\times\) \(\text{4}\)) + (\(\text{7}\) \(\times\) \(\text{3}\) )+ (\(\text{8}\) \(\times\) \(\text{6}\)) + (\(\text{9}\) \(\times\) \(\text{3}\)) + (\(\text{10}\) \(\times\) \(\text{2}\)) = \(\text{148}\)
So the mean is \(\frac{\text{148}}{\text{20}} = \text{7,4}\)

The heights, in centimetres, of boys in the first soccer team are: \(\text{175}\); \(\text{168}\); \(\text{175}\); \(\text{176}\); \(\text{173}\); \(\text{168}\); \(\text{169}\); \(\text{176}\); \(\text{169}\); \(\text{191}\); \(\text{176}\). Find the mean height of these boys.

The frequency table below shows the amount of pocket money, to the nearest Rand that Grade 10 learners are given each week. Calculate the mean amount of pocket money per week.

Pocket money (nearest Rand)

\(\text{30}\)

\(\text{35}\)

\(\text{40}\)

\(\text{45}\)

\(\text{50}\)

Frequency

\(\text{5}\)

\(\text{5}\)

\(\text{10}\)

\(\text{8}\)

\(\text{2}\)

Mean =\(\text{R}\,\text{39,50}\)

For each set of data given in the frequency tables below, find the mean.

Time taken to complete class work (minutes)

\(\text{6}\)

\(\text{9}\)

\(\text{10}\)

\(\text{13}\)

\(\text{15}\)

Frequency

\(\text{4}\)

\(\text{4}\)

\(\text{5}\)

\(\text{4}\)

\(\text{3}\)

Mean = \(\text{10,35}\) minutes

Age of learners (in years)

\(\text{14}\)

\(\text{15}\)

\(\text{16}\)

\(\text{17}\)

\(\text{18}\)

Frequency

\(\text{2}\)

\(\text{3}\)

\(\text{10}\)

\(\text{15}\)

\(\text{10}\)

Mean = \(\text{16,7}\) years

Median

When data is arranged in ascending order, it is arranged from the smallest value to the biggest value (e.g. \(\text{2}\); \(\text{3}\); \(\text{4}\); \(\text{5}\); \(\text{6}\)). When data is arranged in descending order, it is arranged from the biggest value to the smallest value (e.g. \(\text{6}\); \(\text{5}\); \(\text{4}\); \(\text{3}\); \(\text{2}\)). The middle value in the set of data values is called the median. E.g. If you have numbers \(\text{2}\); \(\text{3}\); \(\text{4}\); \(\text{5}\); \(\text{6}\); \(\text{7}\), and \(\text{8}\), the median is \(\text{5}\).

Mode

The mode is the data value that appears most often in a set of data. No calculation is needed to find or determine the mode. You just find the value that appears most frequently. E.g. If you have numbers \(\text{2}\); \(\text{5}\); \(\text{7}\); \(\text{7}\); \(\text{10}\); \(\text{12}\); \(\text{15}\), the mode is \(\text{7}\). If no number is repeated, then there is no mode for the list. You must also be aware that there can be more than one mode.

For grouped data, we use the modal class. This is the group or class that has the highest frequency.

Worked example 5: Finding the mode

Funeka decides to record the colours of schoolbags of everyone arriving at school. She writes down: \(\text{27}\) blue, \(\text{16}\) red, \(\text{43}\) white, \(\text{7}\) black, \(\text{16}\) green. What is the modal colour?

The value \(\text{4}\) appears most often, therefore the mode is \(\text{4}\).

The modal colour is white.

Range

The range is a measure of spread because it tells you how spread out the data values are. The range is found by finding the difference between the largest value and the smallest value.

The heights, in centimetres, of the girls in the cross-country running team are \(\text{175}\); \(\text{168}\); \(\text{175}\); \(\text{176}\); \(\text{173}\); \(\text{168}\); \(\text{169}\); \(\text{176}\); \(\text{169}\); \(\text{191}\); \(\text{176}\) \(\text{cm}\). Find the mean, mode, median and range of the height of these girls.

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